Statistics All-in-One For Dummies

Deborah Rumsey, PhD, is a Professor of Statistics and Statistics Education Specialist at The Ohio State University. She is the author of Statistics For Dummies, Statistics Workbook For Dummies, Statistics II For Dummies, and Probability For Dummies.

Introduction 1

About This Book 1

Foolish Assumptions 2

Icons Used in This Book 3

Beyond the Book 3

Where to Go from Here 4

Unit 1: Getting Started with Statistics 5

Chapter 1: The Statistics of Everyday Life 7

Statistics and the Media: More Questions than Answers? 8

Probing popcorn problems 8

Venturing into viruses 8

Comprehending crashes 9

Mulling malpractice 9

Belaboring the loss of land 10

Scrutinizing schools 10

Scanning sports 10

Banking on business news 11

Touring the travel news 12

Surveying sexual stats 12

Breaking down weather reports 13

Using Statistics at Work 13

Delivering babies — and information 13

Posing for pictures 14

Poking through pizza data 14

Statistics in the office 14

Chapter 2: Taking Control: So Many Numbers, So Little Time 15

Detecting Errors, Exaggerations, and Just Plain Lies 16

Checking the math 16

Uncovering misleading statistics 17

Looking for lies in all the right places 22

Feeling the Impact of Misleading Statistics 23

Chapter 3: Tools of the Trade 25

Thriving in a Statistical World 25

Statistics: More than Just Numbers 26

Designing Appropriate Studies 28

Surveys (Polls) 28

Experiments 29

Collecting Quality Data 30

Sample, random, or otherwise 30

Bias 32

Grabbing Some Basic Statistical Jargon 33

Data 33

Data set 34

Variable 34

Population 34

Statistic 35

Parameter 35

Mean (Average) 36

Median 36

Standard deviation 36

Percentile 37

Standard score 37

Distribution and normal distribution 38

Central Limit Theorem 38

z-values 39

Margin of error 39

Confidence interval 40

Hypothesis testing 41

p-values 42

Statistical significance 42

Correlation, regression, and two-way tables 43

Drawing Credible Conclusions 44

Reeling in overstated results 44

Questioning claims of cause and effect 44

Becoming a Sleuth, Not a Skeptic 45

Unit 2: Number-Crunching Basics 47

Chapter 4: Crunching Categorical Data 49

Summing Up Data with Descriptive Statistics 49

Crunching Categorical Data: Tables and Percents 50

Counting on the frequency 50

Relating with percentages 52

Two-way tables: Summarizing multiple measures 54

Interpreting counts and percents with caution 55

Practice Questions Answers and Explanations 57

Whaddya Know? Chapter 4 Quiz 60

Answers to Chapter 4 Quiz 62

Chapter 5: Means, Medians, and More 65

Measuring the Center with Mean and Median 65

Averaging out to the mean 66

Splitting your data down the median 68

Comparing means and medians: Histograms 70

Accounting for Variation 74

Reporting the standard deviation 75

Being out of range 78

Examining the Empirical Rule (68-95-99.7) 79

Measuring Relative Standing with Percentiles 83

Calculating percentiles 83

Interpreting percentiles 85

Gathering a five-number summary 88

Exploring interquartile range 89

Practice Questions Answers and Explanations 90

Whaddya Know? Chapter 5 Quiz 97

Answers to Chapter 5 Quiz 98

Chapter 6: Getting the Picture: Graphing Categorical Data 99

Take Another Little Piece of My Pie Chart 100

Tallying personal expenses 100

Bringing in a lotto revenue 101

Ordering takeout 102

Projecting age trends 103

Raising the Bar on Bar Graphs 108

Tracking transportation expenses 108

Making a lotto profit 110

Tipping the scales on a bar graph 110

Pondering pet peeves 111

Practice Questions Answers and Explanations 117

Whaddya Know? Chapter 6 Quiz 123

Answers to Chapter 6 Quiz 124

Chapter 7: Going by the Numbers: Graphing Numerical Data 125

Handling Histograms 126

Making a histogram 126

Interpreting a histogram 130

Putting numbers with pictures 137

Detecting misleading histograms 139

Examining Boxplots 143

Making a boxplot 143

Interpreting a boxplot 146

Tackling Time Charts 152

Interpreting time charts 152

Understanding variability: Time charts versus histograms 153

Spotting misleading time charts 153

Practice Questions Answers and Explanations 159

Whaddya Know? Chapter 7 Quiz 165

Answers to Chapter 7 Quiz 166

Unit 3: Distributions and the Central Limit Theorem 167

Chapter 8: Coming to Terms with Probability 169

A Set Notation Overview 169

Noting outcomes: Sample spaces 170

Noting subsets of sample spaces: Events 171

Noting a void in the set: Empty sets 172

Putting sets together: Unions, intersections, and complements 172

Probabilities of Events Involving A and/or B 173

Probability notation 174

Marginal probabilities 175

Union probabilities 175

Intersection ( joint) probabilities 175

Complement probabilities 176

Conditional probabilities 176

Understanding and Applying the Rules of Probability 178

The complement rule (for opposites, not for flattering a date) 178

The multiplication rule (for intersections, not for rabbits) 179

The addition rule (for unions of the nonmarital nature) 180

Recognizing Independence in Multiple Events 182

Checking independence for two events with the definition 182

Using the multiplication rule for independent events 183

Including Mutually Exclusive Events 184

Recognizing mutually exclusive events 184

Simplifying the addition rule with mutually exclusive events 185

Distinguishing Independent from Mutually Exclusive Events 186

Comparing and contrasting independence and exclusivity 186

Checking for independence or exclusivity in a 52-card deck 187

Avoiding Probability Misconceptions 189

Predictions Using Probability 190

Practice Questions Answers and Explanations 192

Whaddya Know? Chapter 8 Quiz 196

Answers to Chapter 8 Quiz 197

Chapter 9: Random Variables and the Binomial Distribution 199

Defining a Random Variable 200

Discrete versus continuous 200

Probability distributions 202

The mean and variance of a discrete random variable 202

Identifying a Binomial 203

Checking binomial conditions step by step 204

No fixed number of trials 204

More than success or failure 205

Trials are not independent 205

Probability of success (p) changes 205

Finding Binomial Probabilities Using a Formula 207

Finding Probabilities Using the Binomial Table 210

Finding probabilities for specific values of X 210

Finding probabilities for X greater-than, less-than, or between two values 211

Checking Out the Mean and Standard Deviation of the Binomial 212

Practice Questions Answers and Explanations 214

Whaddya Know? Chapter 9 Quiz 216

Answers to Chapter 9 Quiz 217

Chapter 10: The Normal Distribution 219

Exploring the Basics of the Normal Distribution 219

Meeting the Standard Normal (Z-) Distribution 223

Checking out Z 223

Standardizing from X to Z 223

Finding probabilities for Z with the Z-table 225

Finding Probabilities for a Normal Distribution 227

Knowing Where You Stand with Percentiles 230

Finding X When You Know the Percent 232

Figuring out a percentile for a normal distribution 232

Translating tricky wording in percentile problems 234

Normal Approximation to the Binomial 236

Practice Questions Answers and Explanations 239

Whaddya Know? Chapter 10 Quiz 248

Answers to Chapter 10 Quiz 249

Chapter 11: The t-Distribution 251

Basics of the t-Distribution 251

Comparing the t- and Z-distributions 251

Discovering the effect of variability on t-distributions 252

Using the t-Table 253

Finding probabilities with the t-table 253

Figuring percentiles for the t-distribution 256

Picking out t*-values for confidence intervals 257

Studying Behavior Using the t-Table 258

Practice Questions Answers and Explanations 259

Whaddya Know? Chapter 11 Quiz 260

Answers to Chapter 11 Quiz 261

Chapter 12: Sampling Distributions and the Central Limit Theorem 263

Defining a Sampling Distribution 264

The Mean of a Sampling Distribution 265

Measuring Standard Error 266

Sample size and standard error 266

Population standard deviation and standard error 267

Looking at the Shape of a Sampling Distribution 269

Case 1: The distribution of X is normal 269

Case 2: The distribution of X is not normal — Enter the Central Limit Theorem 269

Finding Probabilities for the Sample Mean 273

The Sampling Distribution of the Sample Proportion 275

Finding Probabilities for the Sample Proportion 278

Practice Questions Answers and Explanations 280

Whaddya Know? Chapter 12 Quiz 283

Answers to Chapter 12 Quiz 284

Unit 4: Guesstimating and Hypothesizing with Confidence 285

Chapter 13: Leaving Room for a Margin of Error 287

Seeing the Importance of that Plus or Minus 287

Finding the Margin of Error: A General Formula 289

Measuring sample variability 289

Calculating margin of error for a sample proportion 291

Reporting results 293

Calculating margin of error for a sample mean 293

Being confident you’re right 294

Determining the Impact of Sample Size 296

Sample size and margin of error 296

Bigger isn’t always (that much) better! 297

Keeping margin of error in perspective 297

Practice Questions Answers and Explanations 300

Whaddya Know? Chapter 13 Quiz 303

Answers to Chapter 13 Quiz 304

Chapter 14: Confidence Intervals: Making Your Best Guesstimate 305

Not All Estimates Are Created Equal 306

Linking a Statistic to a Parameter 306

Getting with the Jargon 307

Interpreting Results with Confidence 308

Zooming In on Width 310

Choosing a Confidence Level 310

Factoring In the Sample Size 312

Counting On Population Variability 313

Calculating a Confidence Interval for a Population Mean 315

Case 1: Population standard deviation is known 315

Case 2: Population standard deviation is unknown and/or n is small 316

Figuring Out What Sample Size You Need 318

Determining the Confidence Interval for One Population Proportion 319

Creating a Confidence Interval for the Difference of Two Means 322

Case 1: Population standard deviations are known 322

Case 2: Population standard deviations are unknown and/or sample sizes are small 323

Estimating the Difference of Two Proportions 326

Spotting Misleading Confidence Intervals 329

Practice Questions Answers and Explanations 330

Whaddya Know? Chapter 14 Quiz 337

Answers to Chapter 14 Quiz 339

Chapter 15: Claims, Tests, and Conclusions 341

Setting Up the Hypotheses 342

Defining the null 342

What’s the alternative? 343

Gathering Good Evidence (Data) 343

Compiling the Evidence: The Test Statistic 344

Gathering sample statistics 344

Measuring variability using standard errors 344

Understanding standard scores 345

Calculating and interpreting the test statistic 345

Weighing the Evidence and Making Decisions: p-Values 346

Connecting test statistics and p-values 346

Defining a p-value 347

Calculating a p-value 347

Making Conclusions 349

Setting boundaries for rejecting H0 350

Testing varicose veins 350

Assessing the Chance of a Wrong Decision 352

Making a false alarm: Type I errors 353

Missing out on a detection: Type II errors 353

Practice Questions Answers and Explanations 356

Whaddya Know? Chapter 15 Quiz 359

Answers to Chapter 15 Quiz 360

Chapter 16: Commonly Used Hypothesis Tests: Formulas and Examples 361

Testing One Population Mean 362

Handling Small Samples and Unknown Standard Deviations: The t-Test 363

Putting the t-test to work 364

Relating t to Z 365

Handling negative t-values 365

Examining the not-equal-to alternative 365

Drawing conclusions using the critical value 366

Testing One Population Proportion 368

Comparing Two (Independent) Population Averages 371

Case 1: Difference of two population means when population standard deviations are known 371

Case 2: Difference of two population means when population standard deviations are unknown 374

Testing for an Average Difference (The Paired t-Test) 375

Comparing Two Population Proportions 378

Practice Questions Answers and Explanations 382

Whaddya Know? Chapter 16 Quiz 387

Answers to Chapter 16 Quiz 388

Unit 5: Statistical Studies and the Hunt for a Meaningful Relationship 389

Chapter 17: Polls, Polls, and More Polls 391

Recognizing the Impact of Polls 392

Getting to the source 392

Surveying what’s hot 393

Impacting lives 394

Behind the Scenes: The Ins and Outs of Surveys 395

Planning and designing a survey 396

Selecting the sample 399

Carrying out a survey 402

Interpreting results and finding problems 405

Practice Questions Answers and Explanations 408

Whaddya Know? Chapter 17 Quiz 411

Answers to Chapter 17 Quiz 412

Chapter 18: Experiments and Observational Studies: Medical Breakthroughs or Misleading Results? 413

Boiling Down the Basics of Studies 414

Looking at the lingo of studies 414

Observing observational studies 415

Examining experiments 415

Designing a Good Experiment 417

Designing the experiment to make comparisons 417

Selecting the sample size 419

Choosing the subjects 421

Making random assignments 421

Controlling for confounding variables 422

Respecting ethical issues 423

Collecting good data 424

Analyzing the data properly 425

Interpreting Experiment Results 428

Making appropriate conclusions 428

Making informed decisions 429

Practice Questions Answers and Explanations 431

Whaddya Know? Chapter 18 Quiz 433

Answers to Chapter 18 Quiz 434

Chapter 19: Looking for Links: Correlation and Regression 435

Picturing a Relationship with a Scatterplot 436

Making a scatterplot 436

Interpreting a scatterplot 437

Quantifying Linear Relationships Using the Correlation 440

Calculating the correlation 440

Interpreting the correlation 441

Examining properties of the correlation 443

Working with Linear Regression 445

Figuring out which variable is X and which is Y 445

Checking the conditions 446

Calculating the regression line 446

Interpreting the regression line 449

Putting it all together: The regression line for the crickets 451

Making Proper Predictions 453

Checking the conditions 453

Staying in-bounds 454

Regression Analysis: Understanding the Output 456

Residing with Residuals 457

Explaining the Relationship: Correlation versus Cause and Effect 458

Practice Questions Answers and Explanations 460

Whaddya Know? Chapter 19 Quiz 465

Answers to Chapter 19 Quiz 466

Chapter 20: Two-Way Tables and Independence 467

Organizing a Two-Way Table 468

Setting up the cells 469

Figuring the totals 469

Interpreting Two-Way Tables 472

Singling out variables with marginal distributions 472

Examining all groups — a joint distribution 476

Comparing groups with conditional distributions 478

Checking Independence and Describing Dependence 483

Checking for independence 484

Describing a dependent relationship 486

Cautiously Interpreting Results 489

Checking for legitimate cause and effect 489

Projecting from sample to population 490

Making prudent predictions 491

Resisting the urge to jump to conclusions 491

Practice Questions Answers and Explanations 492

Whaddya Know? Chapter 20 Quiz 498

Answers to Chapter 20 Quiz 500

Appendix: Tables for Reference 503

The Z-Table 503

The t-Table 506

The Binomial Table 508

Index 513